MATH Section 1.5

Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃x1∃x2∀z((x1 ≠ x2) ∧ (C(x1, z) ↔ C(x2, z)))

There exist two distinct people enrolled in exactly the same class.

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. No one can fool both Fred and Jerry.

¬∃x(F(x, Fred) ∧ F(x, Jerry))

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. No one can fool himself or herself.

¬∃xF(x, x)

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everybody can fool Fred.

∀xF(x,Fred)

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everybody can fool somebody.

∀x∃yF(x, y)

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Evelyn can fool everybody.

∀yF(Evelyn, y)

Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬∃y ∃xP(x, y)

∀y∀x¬P(x, y)

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements. Everyone can be fooled by somebody.

∀y∃xF(x, y)

Let the domain of discourse be the set of all students in your class. If V(x, y) means x has visited state y, express the statement "Some student in this class has visited Alaska but has not visited Hawaii" using quantifiers.

∃x(V(x, Alaska) ∧¬V(x, Hawaii))

Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. C(Randy Goldberg, CS 252)

Randy Goldberg is enrolled in CS 252.

Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃xC(x, Math 695)

Someone is enrolled in Math 695.

Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. ∃yC(Carol Sitea, y)

Carol Sitea is enrolled in some class.

Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). ¬∀x∃yP(x, y)

∃x∀y ¬P(x, y)

Let the domain of discourse be the set of all students in your class. If G(x, y) means persons x and y grew up in the same town, express the statement "Some student in this class grew up in the same town as exactly one other student in this class" using quantifiers.

∃x∃y(x ≠ y ∧ G(x, y) ∧ ∀z(G(x, z) → (x = y ∨ x = z)))